38
3 Basics of Nonhydrostatic Modelling
plus the addition of a higher-order term according to:
B
+
e = B
n
k + 0.5Ψ
r
+
k
1 − C
+
e
B
n
k+1 − B
n
k
B
−
e = B
n
k+1 − 0.5Ψ
r
−
k
1 + C
−
e
B
n
k+1 − B
n
k
B
+
w = B
n
k−1 + 0.5Ψ
r
+
k−1
1 − C
+
w
B
n
k − B
n
k−1
B
−
w = B
n
k − 0.5Ψ
r
−
k−1
1 + C
−
w
B
n
k − B
n
k−1
where the r parameters are defined by:
r
+
k =
B
n
k − B
n
k−1
B
n
k+1 − B
n
k
and r
−
k =
B
n
k+2 − B
n
k+1
B
n
k+1 − B
n
k
Here, we use the so-called Superbee scheme in which the limiting function Ψ is
defined by:
Ψ(r ) = max {0, min(2r, 1), min(r, 2)}
K¨ ampf (2009) shows performance tests of other limiting functions. Vertical
advection is discretised via fluxes of B through vertical faces of the control volume
in a similar fashion (not shown here). The last term in (3.41), although it should
be zero in theory, is included in this scheme to avoid accumulation of small but
persistent round-off errors that could lead to artificial and unwanted internal sources
or sinks of B.
3.6.3 Stability Criterion for the Advection Equation
The stability criterion for the above explicit form of the advection equation is
given by:
Δt ≤ min
Δx
u
,
Δz
w
(3.43)
known as Courant-Friedrichs-Lewy condition or CFL condition for explicit advection schemes (Courant et al., 1928).
3.6.4 Implementation of Density Diffusion
For uniform grid spacings, the explicit finite-difference versions of the density diffusion terms in Eq. (3.38) are given by:
∂
∂ x
K h
∂ρ
∂ x
=
K
e
h
ρ i,k+1 − ρ i,k
− K
w
h
ρ i,k − ρ i,k−1
/ (Δx)
2
∂
∂z
K z
∂ρ
∂z
=
K
+
z
ρ i−1,k − ρ i,k
− K
−
z
ρ i,k − ρ i+1,k
/ (Δz)
2